Performance of the ARA Transform in Hyperbolic Differential Equations: A Comparative Analysis of First and Second Orders for the Wave Equation

Öz

This study systematically examines the performance of the ARA transform method in solving hyperbolic differential equations, with a specific focus on the wave equation. First and second-order ARA formulations are applied to a one-dimensional wave equation model subject to standard initial conditions. The method's principal feature is its capacity to naturally integrate initial conditions into the solution process while reducing differential equations to algebraic form. The first-order ARA transform provides computational simplicity and ease of use, stemming from its direct relationship with the Laplace transform. In contrast, the second-order formulation offers enhanced capability for addressing more complex differential terms and greater flexibility in incorporating initial conditions. Analytical solutions derived via the ARA transform show complete agreement with the classical D’Alembert solution for both orders. The reliability of the method is confirmed through numerical validation, where the discrepancy between reference solutions and ARA-based results remains within machine precision levels. These findings indicate that the ARA transform achieves high accuracy and theoretical consistency in linear hyperbolic problems. When compared to conventional methods, it proves to be an efficient alternative both theoretically and computationally. A major contribution of this work is highlighting the potential of the method’s parametric nature to be adapted to more challenging problems such as nonlinear systems and fractional differential equations. In this respect, the present study lays a robust foundation for future research and reinforces the position of the ARA transform within the suite of analytical methods.

Anahtar Kelimeler

• Hyperbolic Differential Equations
• Wave Equation
• ARA Transform
• Laplace Transform
• • D'Alembert's Solution

Hiperbolik Diferansiyel Denklemlerde ARA Dönüşümünün Performansı: Dalga Denklemi için Birinci ve İkinci Mertebelerin Karşılaştırmalı Analizi

Öz

Bu çalışma, ARA dönüşüm yönteminin hiperbolik diferansiyel denklemler sınıfındaki dalga denkleminin çözümündeki performansını sistematik bir şekilde incelemektedir. Birinci ve ikinci mertebe ARA formülasyonları, standart başlangıç koşullarına sahip tek boyutlu bir dalga denklemi modeli üzerinde uygulanmıştır. Yöntemin dikkat çeken özelliği, diferansiyel denklemleri cebirsel forma dönüştürürken başlangıç koşullarını çözüme doğal olarak ilave edebilmesidir. Birinci mertebe ARA dönüşümü, Laplace dönüşümü ile kurduğu doğrudan ilişki sayesinde hesaplama kolaylığı ve uygulama basitliği sunmaktadır. İkinci mertebe formülasyon ise daha karmaşık diferansiyel ifadeleri ele alabilme kapasitesi ve başlangıç koşullarını işlemedeki üstün esnekliği ile öne çıkmaktadır. ARA dönüşümden elde edilen analitik çözümlerin, her iki mertebe için de, klasik D'Alembert çözümü ile tam uyum içinde olduğu gözlenmiştir. Yöntemin güvenilirliği, gerçekleştirilen sayısal doğrulama çalışması ile test edilmiş olup, referans çözüm ile ARA yöntemi sonuçları arasındaki farkın makine hassasiyeti seviyelerinde kaldığı belgelenmiştir. Elde edilen bulgular, ARA dönüşümünün doğrusal hiperbolik problemlerde yüksek doğruluk ve teorik tutarlılık sergilediğini ortaya koymaktadır. Geleneksel yöntemlerle karşılaştırıldığında hem teorik hem de hesaplama açısından verimli bir alternatif oluşturduğu sonucuna varılmıştır. Çalışmanın önemli bir katkısı da yöntemin parametrik yapısının, lineer olmayan sistemler ve kesirli diferansiyel denklemler gibi daha karmaşık problemlere uyarlanabilme potansiyelini göstermesidir. Bu açıdan mevcut araştırma, gelecek çalışmalar için sağlam bir temel oluşturmakta ve ARA dönüşümünün analitik yöntemler arasındaki yerini güçlendirmektedir.

Anahtar Kelimeler

• Hiperbolik Diferansiyel Denklemler
• Dalga Denklemi
• ARA Dönüşümü
• D’Alembert Çözümü
• Laplace Dönüşümü

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